g02cg performs a multiple linear regression on a set of variables whose means, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients are given.

Syntax

C#
public static void g02cg( int n, int k1, int k, double[] xbar, double[,] ssp, double[,] r, double[] result, double[,] coef, double[] con, double[,] rinv, double[,] c, out int ifail )

Visual Basic
Public Shared Sub g02cg ( _ n As Integer, _ k1 As Integer, _ k As Integer, _ xbar As Double(), _ ssp As Double(,), _ r As Double(,), _ result As Double(), _ coef As Double(,), _ con As Double(), _ rinv As Double(,), _ c As Double(,), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g02cg ( _
	n As Integer, _
	k1 As Integer, _
	k As Integer, _
	xbar As Double(), _
	ssp As Double(,), _
	r As Double(,), _
	result As Double(), _
	coef As Double(,), _
	con As Double(), _
	rinv As Double(,), _
	c As Double(,), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g02cg( int n, int k1, int k, array<double>^ xbar, array<double,2>^ ssp, array<double,2>^ r, array<double>^ result, array<double,2>^ coef, array<double>^ con, array<double,2>^ rinv, array<double,2>^ c, [OutAttribute] int% ifail )

Visual C++

public:
static void g02cg(
	int n, 
	int k1, 
	int k, 
	array<double>^ xbar, 
	array<double,2>^ ssp, 
	array<double,2>^ r, 
	array<double>^ result, 
	array<double,2>^ coef, 
	array<double>^ con, 
	array<double,2>^ rinv, 
	array<double,2>^ c, 
	[OutAttribute] int% ifail
)

F#
static member g02cg : n : int * k1 : int * k : int * xbar : float[] * ssp : float[,] * r : float[,] * result : float[] * coef : float[,] * con : float[] * rinv : float[,] * c : float[,] * ifail : int byref -> unit

static member g02cg : 
        n : int * 
        k1 : int * 
        k : int * 
        xbar : float[] * 
        ssp : float[,] * 
        r : float[,] * 
        result : float[] * 
        coef : float[,] * 
        con : float[] * 
        rinv : float[,] * 
        c : float[,] * 
        ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: the number of cases $n$ , used in calculating the sums of squares and cross-products and correlation coefficients.

k1: Type: System..::..Int32
On entry: the total number of variables, independent and dependent, $(k + 1)$ , in the regression.

Constraint: $2 \leq k1 < n$ .

k: Type: System..::..Int32
On entry: the number of independent variables $k$ in the regression.

Constraint: $k = k1 - 1$ .

xbar: Type: array<System..::..Double>[]()[][]
An array of size [k1]
On entry: $xbar [i - 1]$ must be set to ${\bar{x}}_{i}$ , the mean value of the $i$ th variable, for $i = 1, 2, \dots, k + 1$ ; the mean of the dependent variable must be contained in $xbar [k]$ .

ssp: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, k1]
Note: dim1 must satisfy the constraint: $dim1 \geq k1$
On entry: $ssp [i - 1, j - 1]$ must be set to $S_{i j}$ , the sum of cross-products of deviations from means for the $i$ th and $j$ th variables, for $i = 1, 2, \dots, k + 1$ and $j = 1, 2, \dots, k + 1$ ; terms involving the dependent variable appear in row $k + 1$ and column $k + 1$ .

r: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, k1]
Note: dim1 must satisfy the constraint: $dim1 \geq k1$
On entry: $r [i - 1, j - 1]$ must be set to $R_{i j}$ , the Pearson product-moment correlation coefficient for the $i$ th and $j$ th variables, for $i = 1, 2, \dots, k + 1$ and $j = 1, 2, \dots, k + 1$ ; terms involving the dependent variable appear in row $k + 1$ and column $k + 1$ .

result

Type: array<System..::..Double>[]()[][]

An array of size [

13

]

On exit: the following information:

$result [0]$	$S S R$ , the sum of squares attributable to the regression;
$result [1]$	$D F R$ , the degrees of freedom attributable to the regression;
$result [2]$	$M S R$ , the mean square attributable to the regression;
$result [3]$	$F$ , the $F$ value for the analysis of variance;
$result [4]$	$S S D$ , the sum of squares of deviations about the regression;
$result [5]$	$D F D$ , the degrees of freedom of deviations about the regression;
$result [6]$	$M S D$ , the mean square of deviations about the regression;
$result [7]$	$S S T$ , the total sum of squares;
$result [8]$	$D F T$ , the total degrees of freedom;
$result [9]$	$s$ , the standard error estimate;
$result [10]$	$R$ , the coefficient of multiple correlation;
$result [11]$	$R^{2}$ , the coefficient of multiple determination;
$result [12]$	${\bar{R}}^{2}$ , the coefficient of multiple determination corrected for the degrees of freedom.

coef

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1,

3

]

Note: dim1 must satisfy the constraint:

dim1 \geq k

On exit: for

i = 1, 2, \dots, k

, the following information:

$coef [i - 1, 0]$: $b_{i}$ , the regression coefficient for the $i$ th variable.
$coef [i - 1, 1]$: $s e (b_{i})$ , the standard error of the regression coefficient for the $i$ th variable.
$coef [i - 1, 2]$: $t (b_{i})$ , the $t$ value of the regression coefficient for the $i$ th variable.

con

Type: array<System..::..Double>[]()[][]

An array of size [

3

]

On exit: the following information:

$con [0]$	$a$ , the regression constant;
$con [1]$	$s e (a)$ , the standard error of the regression constant;
$con [2]$	$t (a)$ , the $t$ value for the regression constant.

rinv: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, k]
Note: dim1 must satisfy the constraint: $dim1 \geq k$
On exit: the inverse of the matrix of correlation coefficients for the independent variables; that is, the inverse of the matrix consisting of the first $k$ rows and columns of r.

c

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, k]

Note: dim1 must satisfy the constraint:

dim1 \geq k

On exit: the modified inverse matrix, where

$c [i - 1, j - 1] = r [i - 1, j - 1] \times rinv [i - 1, j - 1] / ssp [i - 1, j - 1]$ , for $i = 1, 2, \dots, k$ and $j = 1, 2, \dots, k$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

g02cg fits a curve of the form

y = a + b_{1} x_{1} + b_{2} x_{2} + \dots + b_{k} x_{k}

to the data points

\begin{matrix} (x_{11}, x_{21}, \dots, x_{k 1}, y_{1}) \\ (x_{12}, x_{22}, \dots, x_{k 2}, y_{2}) \\ ⋮ \\ (x_{1 n}, x_{2 n}, \dots, x_{k n}, y_{n}) \end{matrix}

such that

y_{i} = a + b_{1} x_{1 i} + b_{2} x_{2 i} + \dots + b_{k} x_{k i} + e_{i}, i = 1, 2, \dots, n .

The method calculates the regression coefficients,

b_{1}, b_{2}, \dots, b_{k}

, the regression constant,

a

, and various other statistical quantities by minimizing

\sum_{i = 1}^{n} e_{i}^{2} .

The actual data values

(x_{1 i}, x_{2 i}, \dots, x_{k i}, y_{i})

are not provided as input to the method. Instead, input consists of:

(i)	The number of cases, $n$ , on which the regression is based.
(ii)	The total number of variables, dependent and independent, in the regression, $(k + 1)$ .
(iii)	The number of independent variables in the regression, $k$ .
(iv)	The means of all $k + 1$ variables in the regression, both the independent variables $(x_{1}, x_{2}, \dots, x_{k})$ and the dependent variable $(y)$ , which is the $(k + 1)$ th variable: i.e., ${\bar{x}}_{1}, {\bar{x}}_{2}, \dots, {\bar{x}}_{k}, \bar{y}$ .
(v)	The $(k + 1)$ by $(k + 1)$ matrix [ $S_{i j}$ ] of sums of squares and cross-products of deviations from means of all the variables in the regression; the terms involving the dependent variable, $y$ , appear in the $(k + 1)$ th row and column.
(vi)	The $(k + 1)$ by $(k + 1)$ matrix [ $R_{i j}$ ] of the Pearson product-moment correlation coefficients for all the variables in the regression; the correlations involving the dependent variable, $y$ , appear in the $(k + 1)$ th row and column.

The quantities calculated are:

(a)

The inverse of the

k

k

partition of the matrix of correlation coefficients, [

R_{i j}

], involving only the independent variables. The inverse is obtained using an accurate method which assumes that this sub-matrix is positive definite.

(b)

The modified inverse matrix,

C = [c_{i j}]

, where

c_{i j} = \frac{R_{i j} r_{i j}}{S_{i j}}, i, j = 1, 2, \dots, k,

where

r_{i j}

is the

(i, j)

th element of the inverse matrix of [

R_{i j}

] as described in (a) above. Each element of

C

is thus the corresponding element of the matrix of correlation coefficients multiplied by the corresponding element of the inverse of this matrix, divided by the corresponding element of the matrix of sums of squares and cross-products of deviations from means.

(c)

The regression coefficients:

b_{i} = \sum_{j = i}^{k} c_{i j} S_{j (k + 1)}, i = 1, 2, \dots, k,

where

S_{j (k + 1)}

is the sum of cross-products of deviations from means for the independent variable

x_{j}

and the dependent variable

y

(d)

The sum of squares attributable to the regression,

S S R

, the sum of squares of deviations about the regression,

S S D

, and the total sum of squares,

S S T

$S S T = S_{(k + 1) (k + 1)}$ , the sum of squares of deviations from the mean for the dependent variable, $y$ ;
$S S R = \sum_{j = 1}^{k} b_{j} S_{j (k + 1)}; S S D = S S T - S S R$

(e)

The degrees of freedom attributable to the regression,

D F R

, the degrees of freedom of deviations about the regression,

D F D

, and the total degrees of freedom,

D F T

D F R = k; D F D = n - k - 1; D F T = n - 1 .

(f)

The mean square attributable to the regression,

M S R

, and the mean square of deviations about the regression,

M S D

M S R = S S R / D F R; M S D = S S D / D F D .

(g)

The

F

values for the analysis of variance:

F = M S R / M S D .

(h)

The standard error estimate:

s = \sqrt{M S D} .

(i)

The coefficient of multiple correlation,

R

, the coefficient of multiple determination,

R^{2}

and the coefficient of multiple determination corrected for the degrees of freedom,

{\bar{R}}^{2}

;

R = \sqrt{1 - \frac{S S D}{S S T}}; R^{2} = 1 - \frac{S S D}{S S T}; {\bar{R}}^{2} = 1 - \frac{S S D \times D F T}{S S T \times D F D} .

(j)

The standard error of the regression coefficients:

s e (b_{i}) = \sqrt{M S D \times c_{i i}}, i = 1, 2, \dots, k .

(k)

The

t

values for the regression coefficients:

t (b_{i}) = \frac{b_{i}}{s e (b_{i})}, i = 1, 2, \dots, k .

(l)

The regression constant,

a

, its standard error,

s e (a)

, and its

t

value,

t (a)

a = \bar{y} - \sum_{i = 1}^{k} b_{i} {\bar{x}}_{i}; s e (a) = \sqrt{M S D \times (\frac{1}{n} + \sum_{i = 1}^{k} \sum_{j = 1}^{k} {\bar{x}}_{i} c_{i j} {\bar{x}}_{j})}; t (a) = \frac{a}{s e (a)} .

References

Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley

Error Indicators and Warnings

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDSSP, LDR, LDCOEF, LDRINV, LDC) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$

On entry,

k1 < 2

$ifail = 2$

On entry,

k1 \neq (k + 1)

$ifail = 3$

On entry,

n \leq k1

$ifail = 4$

On entry,

ldwkz < k

$ifail = 5$: The $k$ by $k$ partition of the matrix $R$ which is to be inverted is not positive definite.

$ifail = 6$: The refinement following the actual inversion fails, indicating that the $k$ by $k$ partition of the matrix $R$ , which is to be inverted, is ill-conditioned. The use of g02da, which employs a different numerical technique, may avoid this difficulty (an extra ‘variable’ representing the constant term must be introduced for g02da).

$ifail = 7$: Unexpected error in (F04ABF not in this release).

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The accuracy of any regression method is almost entirely dependent on the accuracy of the matrix inversion method used. In g02cg, it is the matrix of correlation coefficients rather than that of the sums of squares and cross-products of deviations from means that is inverted; this means that all terms in the matrix for inversion are of a similar order, and reduces the scope for computational error. For details on absolute accuracy, the relevant section of the document describing the inversion method used, (F04ABF not in this release), should be consulted. g02da uses a different method, based on (F04AMF not in this release), and that method may well prove more reliable numerically. It does not handle missing values, nor does it provide the same output as this method. (In particular it is necessary to include explicitly the constant in the regression equation as another ‘variable’.)

If, in calculating

F

t (a)

, or any of the

t (b_{i})

(see [Description]), the numbers involved are such that the result would be outside the range of numbers which can be stored by the machine, then the answer is set to the largest quantity which can be stored as a real variable, by means of a call to x02al.

Parallelism and Performance

None.

Further Comments

The time taken by g02cg depends on

k

This method assumes that the matrix of correlation coefficients for the independent variables in the regression is positive definite; it fails if this is not the case.

This correlation matrix will in fact be positive definite whenever the correlation matrix and the sums of squares and cross-products (of deviations from means) matrix have been formed either without regard to missing values, or by eliminating completely any cases involving missing values, for any variable. If, however, these matrices are formed by eliminating cases with missing values from only those calculations involving the variables for which the values are missing, no such statement can be made, and the correlation matrix may or may not be positive definite. You should be aware of the possible dangers of using correlation matrices formed in this way (see the G02 class), but if they nevertheless wish to carry out regression using such matrices, this method is capable of handling the inversion of such matrices provided they are positive definite.

If a matrix is positive definite, its subsequent re-organisation by either g02ce or g02cf will not affect this property, and the new matrix can safely be used in this method. Thus correlation matrices produced by any of g02ba, g02bb, g02bg or g02bh, even if subsequently modified by either g02ce or g02cf, can be handled by this method.

It should be noted that in forming the sums of squares and cross-products matrix and the correlation matrix a column of constants should not be added to the data as an additional ‘variable’ in order to obtain a constant term in the regression. This method automatically calculates the regression constant,

a

, and any attempt to insert such a ‘dummy variable’ is likely to cause the method to fail.

It should also be noted that the method requires the dependent variable to be the last of the

k + 1

variables whose statistics are provided as input to the method. If this variable is not correctly positioned in the original data, the means, standard deviations, sums of squares and cross-products of deviations from means, and correlation coefficients can be manipulated by using g02ce or g02cf to reorder the variables as necessary.

Example

This example reads in the means, sums of squares and cross-products of deviations from means, and correlation coefficients for three variables. A multiple linear regression is then performed with the third and final variable as the dependent variable. Finally the results are printed.

Example program (C#): g02cge.cs

Example program data: g02cge.d

Example program results: g02cge.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also